We study a class of semilinear SchrSdinger equation with electromagnetic fields and the nonlinearity term involving critical growth. We assume that the potential of the equation includes a parameter A and can be negative in some domain. Moreover, the potential behaves like potential well when the parameter A is large. Using variational methods combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter A becomes large, localized near the bottom of the potential well. Our result is an extension of the corresponding result for the SchrSdinger equation which involves critical growth but does not involve electromagnetic fields.
We are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the fractional Laplacian（-△）%s u（x）＋λV（x）u（x）=u（x）^（p-1）,u（x）〉=0,x∈R^N,for sufficiently large λ,2〈p〈N-2s^-2N for N≥2. V（x） is a real continuous function on RN. Using variational methods we prove the existence of least energy solution uλ（x） which localizes near the potential well int V-1 （0） for A large. Moreover, if the zero sets int V-1 （0） of V（x） include more than one isolated component, then ux（x） will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter A is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V^-1（0）. This is the essential difference with the Laplacian problems since the operator （-△）s is nonlocal.
